Theorem raleqtrdv 2757

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)

Hypotheses

Ref	Expression
raleqtrdv.1	|- ( ph -> A. x e. A ps )
raleqtrdv.2	|- ( ph -> A = B )

Assertion

Ref	Expression
raleqtrdv	|- ( ph -> A. x e. B ps )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem raleqtrdv

Step	Hyp	Ref	Expression
1		raleqtrdv.1	. 2 |- ( ph -> A. x e. A ps )
2		raleqtrdv.2	. . 3 |- ( ph -> A = B )
3	2	raleqdv 2755	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
4	1, 3	mpbid 147	1 |- ( ph -> A. x e. B ps )

Syntax hints:   -> wi 4   = wceq 1402   A.wral 2528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533

This theorem is referenced by:  znf1o  14969  upgr2wlkdc  16601